| L(s) = 1 | + 3-s − 1.41·5-s + 9-s + 2.82·11-s + 4.24·13-s − 1.41·15-s − 4.24·17-s − 8·19-s − 2.82·23-s − 2.99·25-s + 27-s − 4·31-s + 2.82·33-s − 4·37-s + 4.24·39-s − 7.07·41-s + 5.65·43-s − 1.41·45-s + 4·47-s − 4.24·51-s − 6·53-s − 4.00·55-s − 8·57-s − 8·59-s + 4.24·61-s − 6·65-s − 2.82·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.632·5-s + 0.333·9-s + 0.852·11-s + 1.17·13-s − 0.365·15-s − 1.02·17-s − 1.83·19-s − 0.589·23-s − 0.599·25-s + 0.192·27-s − 0.718·31-s + 0.492·33-s − 0.657·37-s + 0.679·39-s − 1.10·41-s + 0.862·43-s − 0.210·45-s + 0.583·47-s − 0.594·51-s − 0.824·53-s − 0.539·55-s − 1.05·57-s − 1.04·59-s + 0.543·61-s − 0.744·65-s − 0.340·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.119511816886509853879346943688, −7.24065729802798303048108195209, −6.49477747856088041096510379518, −5.99172313878680032711579115587, −4.70294722263552824365601185135, −3.93625971165578040017488646154, −3.64934999462054953422919667497, −2.33750585235070308322619363315, −1.54137197897053774247752283515, 0,
1.54137197897053774247752283515, 2.33750585235070308322619363315, 3.64934999462054953422919667497, 3.93625971165578040017488646154, 4.70294722263552824365601185135, 5.99172313878680032711579115587, 6.49477747856088041096510379518, 7.24065729802798303048108195209, 8.119511816886509853879346943688
